1. Physics 3 for Electrical Engineering
Ben Gurion University of the Negev
Physics 3 for Electrical Engineering
Lecturers: Daniel Rohrlich, Ron Folman
Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 10. Quantum mechanics – Schrödinger’s equation for the hydrogen atom • eigenvalues and eigenstates • atomic quantum numbers • Stern-Gerlach and spin • Pauli matrices • spin-orbit coupling
Sources: Feynman Lectures III, Chap. 19 Sects. 1-5;
Merzbacher (2nd edition) Chap. 9;
Merzbacher (3rd edition) Chap. 12;
פרקים בפיסיקה מודרנית, יחידה 8, פרקים 1-2
Tipler and Llewellyn, Chap. 7 Sects. 2-5.

2. Schrödinger’s equation for the hydrogen atom
The most common isotope of hydrogen contains just one proton and one electron. Their potential energy is
so Schrödinger’s equation depends on rp and re , i.e. on six coordinates:
But a clever change of variables makes the equation simpler:

3. Schrödinger’s equation for the hydrogen atom
Let r = re – rp and then Schrödinger’s
equation becomes
where is called the reduced mass. For a proton
and an electron, the reduced mass is essentially the electron mass, since mp ≈ 2000 me. But for a positronium atom (which has a positron in the place of the proton), μ = me /2.

4. Schrödinger’s equation for the hydrogen atom
Let r = re – rp and then Schrödinger’s
equation becomes
where is called the reduced mass. There are
solutions of the form Ψ = T(t)χ(R)ψ(r), where T(t) is the usual time dependence and χ(R) is a solution to the free Schrödinger equation in R (the center-of-mass coordinate), which has no potential term.

5. Schrödinger’s equation for the hydrogen atom
The equation for ψ(r) is then
with the Coulomb energy as a central potential V(r).
We have seen that, for a central potential, Schrödinger’s equation reduces to

6. We replace m by μ and V(r) by
and obtain the eigenvalues of ; they are We solved this equation by expressing ψ(r,θ,φ) as a product of two functions:
ψ(r,θ,φ) = R(r)Ylm(θ,φ) ,
where

7. Eigenvalues and eigenstates
Hence the equation for R(r) is
The solutions Rnl(r) of this equation have the form
with Fnl (r/a0) a polynomial and n ≥ l The constant a0 is called the Bohr radius and equals Å

8. Eigenvalues and eigenstates
The energies depend only on n:
where 1 eV =  × 10-19 J.
Question: What is the normalization condition for ψnlm(r,θ,φ)?

9. Eigenvalues and eigenstates
The energies depend only on n:
where 1 eV =  × 10-19 J.
Question: What is the normalization condition for ψnlm(r,θ,φ)?
Answer:

10. Eigenvalues and eigenstates
Question: What is the normalization condition for Rn0(r)?

11. Eigenvalues and eigenstates
Question: What is the normalization condition for Rn0(r)?
Answer:

12. Eigenvalues and eigenstates
Here are the lowest normalized solutions ψnlm = Rnl(r)Ylm(θ,φ) of the hydrogen atom Schrödinger equation (see also here):
n l m ψnlm

13. Radial eigenfunctions Rnl(r) and probability distributions Pnl(r) for the lowest eigenstates of the hydrogen atom.
From here.
Question: How can Pn0(r) vanish at r = 0 if Rn0(r) does not?
Rnl(r)
Pnl(r)
Radius (a0)

14. Radial eigenfunctions Rnl(r) and probability distributions Pnl(r) for the lowest eigenstates of the hydrogen atom.
From here.
Question: How can Pn0(r) vanish at r = 0 if Rn0(r) does not?
Answer: On a sphere of radius r, we have Pn0(r) = r2 |Rn0(r)|2.
Rnl(r)
Pnl(r)
Radius (a0)

15. For a single electron bound by a nucleus containing Z protons, the solutions of Schrödinger’s equation are almost unchanged; the reduced mass μ is even closer to me, while the potential term is
Thus by replacing e2 with Ze2 in the eigenstates and eigenvalues we obtained for Z = 1, we obtain the Z > 1 eigenstates and eigenvalues.
For example, we obtain the energy eigenvalues

16. Exercise: Show that the energy of the ground state is half the expectation value of the potential energy in the ground state.

17. Exercise: Show that the energy of the ground state is half the expectation value of the potential energy in the ground state.
Solution: The ground state energy is
The expectation value of the potential energy is

18. Atomic quantum numbers
According to what we have seen so far, every eigenstate of the hydrogen atom can be associated with three quantum numbers n, l and m, where
n = 1, 2, 3, … “principle” quantum number
l = 0, …, n – 1;
m = –l, –l +1, …, l – 1, l .
The degeneracy of the energy eigenvalue En is therefore
… + [2(n – 1) + 1] = n2 .
In atomic physics, the l quantum numbers have special names:
s for l = 0, p for l = 1, d for l = 2, f for l = 3, etc. Then 2s means n = 2 and l = 0; 3p means n = 3 and l = 1; and so on.

19. Since the energy of an electron in a stationary hydrogen atom can only be one of the En , where
a stationary atom can absorb and emit photons only if the photon energy equals Ephoton = En – En’ . And since the energy of a photon is related to its frequency ν by Ephoton = hν, the frequencies of electromagnetic radiation emitted or absorbed by a hydrogen atom must obey the rule
for some n and n´. This formula, derived by Bohr 13 years before Schrödinger, was soon verified via spectroscopy.

20. Energy levels for hydrogen, and transitions among the levels

21. Energy levels for hydrogen, and transitions among the levels
Corresponding spectral lines

22. Setup for emission spectroscopy:
Light from source

23. The “magnetic” quantum number m:
When an atom is immersed in a uniform magnetic field, the energies En split! Let B be the strength of the field and let point up the z-axis. A state with quantum numbers n, l, m has energy
where μB = eħ/2me is the Bohr magneton. How can we explain this effect? An electron moving in a circular orbit of radius r, at speed v, produces a current I = ev/2πr and a magnetic moment μz = I(πr2) = evr/2 = eLz/2me. Since Lz = mħ, we have μz = μBm.
The corresponding extra potential term for the hydrogen atom is

24. The “magnetic” quantum number m:
But this is not the only magnetic effect. In a uniform magnetic field, there is a torque on the atom (to make it anti-parallel to the magnetic field). But in a non-uniform magnetic field, there is also a force on the atom. Assume again that the field points up the z-axis, so that VB = –μz B. Then if dB/dz ≠ 0, the force is
So, what happens if a beam of neutral atoms with –lħ ≤ Lz ≤ lħ crosses a non-uniform magnetic field?

25. The “magnetic” quantum number m:
But this is not the only magnetic effect. In a uniform magnetic field, there is a torque on the atom (to make it anti-parallel to the magnetic field). But in a non-uniform magnetic field, there is also a force on the atom. Assume again that the field points up the z-axis, so that VB = –μz B. Then if dB/dz ≠ 0, the force is
So, what happens if a beam of neutral atoms with –lħ ≤ Lz ≤ lħ crosses a non-uniform magnetic field?
We expect the beam to split into 2l+1 beams, one for each value of μz. In some cases, the beam indeed splits into 2l+1 beams.

26. Stern-Gerlach and spin
But O. Stern and W. Gerlach saw a beam of silver atoms split into two beams!

27. Stern-Gerlach and spin
But O. Stern and W. Gerlach saw a beam of silver atoms split into two beams!
How can have an even number of eigenvalues?
G. Uhlenbeck and S. Goudsmit suggested that each electron has its own intrinsic angular momentum – “spin” – with only two eigenvalues.
But electron spin has odd features. For example, its magnitude never changes, just its direction – and it has only two directions.

28. Stern-Gerlach and spin
Let’s try to understand spin better by reviewing the algebra of
Consider l = 1 and m = –1, 0, 1. The
matrix representation of in a basis of eigenstates of is
since the eigenvalues are 0 and ±ħ.

29. Stern-Gerlach and spin
What is ? We know it must equal but what is a?
We have
Hence , up to an overall phase. Similarly, we
can show that , hence

30. Stern-Gerlach and spin
Similarly, Now, since ,
we can write Since
we can write

31. Pauli matrices
It is straightforward to check that these matrix representations
have the correct commutation relations:
But Pauli discovered 2 × 2 matrices with the same commutation relations:
(The “Pauli matrices” are these matrices without the ħ/2 factors.)
These are the operators for the components of electron spin!

32. Pauli matrices
We can write the eigenstates of
as for Sz = ħ/2
and as for Sz = –ħ/2.
Since for s = ½,
we refer to electron spin as “spin-½”.

33. Stern-Gerlach and spin
More odd features of electron spin:
The eigenvalues of are ±ħ/2.
We can write an eigenstate of with eigenvalue mħ as
but an analogous eigenstate of , namely
, would not be single-valued. Yet experiments show
that these electron spin eigenstates are not invariant under rotation by 2π, but they are invariant under rotation by 4π!
This is reminiscent of a trick with a twisted ribbon….

34. Stern-Gerlach and spin
This is reminiscent of a trick with a twisted ribbon…one twist cannot be undone, but two twists are equivalent to no twist.

35. Stern-Gerlach and spin
One more odd feature of electron spin:
For orbital angular momentum, we found that μz = eLz/2me.
For spin angular momentum, experiment shows that μz = eSz/me. That is, electronic spin produces an anomalous “double” magnetic moment.
Therefore, the total magnetic moment of an electron with orbital angular momentum mħ and spin angular momentum ±ħ/2 is

36. Atomic quantum numbers (again)
We associated every eigenstate of the hydrogen atom with three quantum numbers n, l and m. But now we have to introduce a fourth quantum number, the spin: ms = ±½ .
The degeneracy of the energy eigenvalue En is therefore not
n2 but 2n2, since there are two spin states for every set of quantum numbers n, l and m.
The nucleus, too, has spin angular momentum. But its magnetic moment is relatively tiny because the mass of a proton is about 2000 times the electron mass. In this course we neglect the spin and magnetic moment of the nucleus.

37. Exercise: Show that the superposition of wave functions
is normalized if each wave function is, and calculate
and ΔLz.

38. Exercise: Show that the superposition of wave functions
is normalized if each wave function is, and calculate
and ΔLz.
Solution: Since the components have different eigenvalues, they are orthonormal, and the normalization is obtained from the absolute value of the squares of the coefficients:

39. Exercise: Show that the superposition of wave functions
is normalized if each wave function is, and calculate
and ΔLz.
Solution:

40. Exercise: Show that the superposition of wave functions
is normalized if each wave function is, and calculate
and ΔLz.
Solution:

41. Exercise: What happens in a Stern-Gerlach experiment, if each electron in an incident beam of hydrogen atoms has l = 1?

42. Exercise: What happens in a Stern-Gerlach experiment, if each electron in an incident beam of hydrogen atoms has l = 1?
Solution: The magnetic moment of the electron depends on Lz and Sz according to
Since m = –1, 0, 1 and, independently, ms = ±½, we get five possible values of m +2ms : 2, 1, 0, –1, –2. We therefore expect to see 5 separate spots on the screen.

43. Spin-orbit coupling
We discussed atomic magnetic moments in a magnetic field that is uniform or non-uniform. But even without any external magnetic field, an electron feels an effective field. Why?

44. Spin-orbit coupling
We discussed atomic magnetic moments in a magnetic field that is uniform or non-uniform. But even without any external magnetic field, an electron feels an effective field. Why?
The electron moves relative to the nucleus. Transforming the Coulomb field to the electron’s rest frame yields a magnetic field B' = –v × E/c2. Since E is radial, –v × E/c2 is –dV(r)/dr times r × p /emec2r = L/emec2r. Since the electron’s magnetic
moment e /me interacts with B', the spin-orbit interaction contains also It enters the Hamiltonian as
where V(r) is the Coulomb potential.

45. Spin-orbit coupling
To compute the eigenvalues of , we must know how to add
angular momenta. Defining , we find that
and so on, i.e. the components of follow exactly the same algebra as the components of and We immediately infer that the eigenvalues of are j(j+1)ħ2 and that the eigenvalues of are –jħ, (–j+1)ħ,…, (j–1)ħ, jħ.

46. Spin-orbit coupling
To compute the eigenvalues of , we must know how to add
angular momenta. Defining , we find that
and so on, i.e. the components of follow exactly the same algebra as the components of and We immediately infer that the eigenvalues of are j(j+1)ħ2 and that the eigenvalues of are –jħ, (–j+1)ħ,…, (j–1)ħ, jħ.
Now from we derive

47. Exercise: The spin-orbit coupling splits the degeneracy between
the hydrogen states ψ2,1,1, –½ and ψ2,1,1,½ by ΔE = 4.5 × 10-5 eV. Estimate the magnetic field B' felt by the electron.

48. Exercise: The spin-orbit coupling splits the degeneracy between
the hydrogen states ψ2,1,1, –½ and ψ2,1,1,½ by ΔE = 4.5 × 10-5 eV. Estimate the magnetic field B' felt by the electron.
Solution: The energy splitting is due to the interaction of the electron’s magnetic moment with the effective magnetic field B'. In the rest frame of the electron only the spin magnetic moment contributes: μzB' = (eSz/me)B' = ±eB'ħ/2me, hence ΔE = eB'ħ/me and
B' = meΔE/eħ
= ΔE/2μB
= (4.5 × 10-5 eV) / 2 × (5.79 × 10-5 eV/T)
= 0.39 T .